On Hadamard products of linear varieties

In this paper, we address the Hadamard product of linear varieties not necessarily in general position. In [Formula: see text], we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets [Formula: see text] and [Formula: see text] of collinear points, we get conditions for [Formula: see text] to be either a collinear finite set of points or a grid of [Formula: see text] points. In [Formula: see text] under suitable conditions (which we prove to be generic), we show that [Formula: see text] consists of [Formula: see text] points on the two different rulings of a nondegenerate quadric and we compute its Hilbert function in the case [Formula: see text]

FINITE PROJECTIVE SPACES, GEOMETRIC SPREADS OF LINES AND MULTI-QUBITS

Given a (2N-1)-dimensional projective space over GF(2), PG (2N-1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG (N-1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG (N-1, 4). Under such mapping, a nondegenerate quadric surface of the former space has for its image a nonsingular Hermitian variety in the latter space, this quadric being hyperbolic or elliptic in dependence on N being even or odd, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric Pauli operators into a new perspective. The N = 4 case is taken to illustrate the issue, due to its link with the so-called black-hole/qubit correspondence.

Rational Quadratic Parameterizations of Quadrics

Every irreducible quadric in E3 has infinitely many different rational quadratic parameterizations. These parameterizations and the relationships between them are investigated. It is shown that every faithful rational quadratic parameterization of a quadric can be generated by a stereographic projection from a point on the quadric, called the center of projection (COP). Two such parameterizations for the same quadric are related by a rational linear reparameterization if they have the same COP; otherwise they are related by a rational quadratic reparameterization. We also consider unfaithful parameterizations for which, in general, a one-to-one correspondence between points on the surface and parameters in the plane does not exist. It is shown that all unfaithful rational quadratic parameterizations of a properly degenerate quadric can be characterized by a simple canonical form, and there exist no unfaithful rational quadratic parameterizations for a nondegenerate quadric. In addition, given a faithful rational quadratic parameterization of a quadric, a new technique is presented to compute its base points and inversion formula. These results are applied to solve the problems of parameterizing the intersection of two quadrics and reparameterizing a given quadric parameterization with respect to a different COP without implicitization.